On an inequality for the Smarandache function J. Sandor Baheยง-Bolyai University, 3.JOO Cluj-Napoca. Romania
1. In paper [2J the author proved among others the inequality S(ab) :::; as(b) for all a, b positive integers. This was refined to
S(ab) :::; S(a)
+ S(b)
(1)
in [1 J. Our aim is to shmv that certain results from om recent paper [3] can be obtained in a simpler way from a generalization of relation (1). On the other hand, by the method of Le [lJ we can deduce similar, more complicated inequalities of type (1). 2. By mathematical induction we have from (1) immediately:
(2) for all integers ai 2 1 (i
= L ... , n).
When
al
= ... = an = n
we obtain
(3) For three applications of this inequality. remark that
S((m!t) :::; nS(m!) = nm
(4)
since S( m!) = m. This is inequality 3) part 1. from [3J. By the same way, S( (n!)(n-l)!) :::;
(n - l)!S(n!) = (n - l)!n = n!, i.e.
(5)
125
Inequality (5) has been obtain<~d in [3] by other arguments (see 4) part 1.). Finally, by 5(n 2 ) ~. 25(n) ~ n for n even (see [3], ineqllality 1), n > 4, we have obtained a refinement of 5( n 2 ) ~ n:
(6) for n > 4, even. 3. Let m he a divisor of n. i.e. n
= km. Then (1) gives 5(n) = 5(km)
~
5(m) + 5(k),
so we obtain:
If min, then
5(n) - 5(m)
~ 5 (:) .
(7)
As an application of (7), let d( n) be the number of divisors of n. Since and
IT k = n1 (see [3]), and by IT /.:1 IT k, from (7) we can deduce that k<n
kin
IT k = n
d
(n)/2,
kin
k<n
(8) This improves our relation (10) from [3]. 4. Let 5(a)
But from alu1
= u. 5(b) = v. Then blv1 and u!jx(x-l) ... (x-u+l) for all integers x ~ u. we have alx(x - 1) ... (x - u + 1) for all x ~ u. Let x = u + v + k (k ~ 1).
Then. clearly ab( v+ 1) ... (v+k)l( u +v+k)!' so we have 5[ab( v+ 1) ... (v +k)] ~ u +v+k. Here v = 5(b), so we have obtained that
5[ab(S(b)
+ 1) ... (5(b) + k)]
:::; 5(a) -+- 5(b)
+ k.
(9)
For example, for k = lone has
5[ab(5(b)
+ 1)]
:::; 5(a)
+ 5(b) + 1.
This is not a consequence of (2) for n = 3, since 5[5(b)
126
+ 1]
(10)
may be much larger than 1.
References [1] M. Le, An ineqTuzlity concerning the Smarandache junction, Smarandache Notions J., vol. 9(1998), 124-125. [2] J. Sandor, On certain inequalities involving the Smarandache junction, Smarandache Notions J., vol. 7(1996), 3-6. [3] J. Sandor. On certain new inequalities and limits jor the Smarandache junction, Smarandache Notions J., vol. 9(1998), 63-69.
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